Carrot-and-turnip Gratin

• 1 egg
• 1/2 thinly scallion greens (sliced)
• 475 ml milk
• 180 gr carrots
• 140 gr turnips
• 120 ml cream (heavy)
• 40 gr parmesan (freshly grated)
• 32 gr cornstarch
• 8 gr butter (cold unsalted)
• 3 gr parsley (minced fresh)